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temperature. Bearing designs that incorporate grooves or tilting pads are also effective in inhibiting oil-whirl instability.
Aerodynamic .hirl.Although the mechanism is not clearly understood，it has been shown that aerodynamic components， such as compressor wheelsand turbine wheels， can create cross-coupled forces due to the wheel motion. Figure 5-23b is one representation of how such forces may be induced.
The acceleration or deceleration of the process fluid imparts a net tangen-tial force on the blading. If the clearance between the wheel and housingvaries circumferentially， a variation of the tangential forces on the bladingmay also be expected， resulting in a net destabilizing force as shown in Figure 5-23b. The resultant force from the cross-coupling of angular motion and radial forces may destabilize the rotor and cause a whirl motion.
The aerodynamic cross-coupling effect has been quantified into equivalentstiffness. Forinstance， in axial-flow machines
.T
主xω二 主ωx二 D. (5-42)

Figure 5-2.b. Aerodynamic cross-coupling (Ehrich，F(xiàn) F ， ""Identification and Avoidance of Instabilities and Self-Excited .ibrations inRotatingMachinery，""Adopted from ASME Paper72-DE-21， General ElectricCo ， Aircraft EngineGroup，Group EngineeringDivision， May11， 1972 )
where:
.二 efficiency slope versus displacement over blade-height curve
T二 stage torque
D二 average pitch diameter
.二 average blade height

The stiffness that results from the previous quantification may be used in a critical-speed program in much the same manner as bearing coefficients.
.hirl from fluid trapped in the rotor. This type of whirl occurs when liquids are inadvertently trapped in an internal rotor cavity. The mechanism of this instability is shown in Figure 5-24. The fluid does not flow in a radial direction but flows in a tangential direction. The onset of instability occurs between the first and second critical speeds. Table 5-4 is a handy summary for both avoidance and diagnosis of self-excitation and instabilities in rotat-ing shafts.

.ampbell Diagram
The Campbell diagram is an overall or bird's-eye view of regional vibra-tion excitation that can occur on an operating system. The Campbell dia-gram can be generated from machine design criteria or from machine operating data. A typical Campbell diagram plot is shown in Figure 5-25. Engine rotational speed is along the X axis. The system frequency is along the Yaxis. The fan lines are engine-order lines: one-half engineorder， onetimes engine order， two times engineorder， three times engineorder， fourtimes engineorder， five times engine order， 10 times engine order， etc. Thisform of design study is necessary， especially when designing an axial com-pressor to determine if a natural blade frequency is excited by a runningfrequency， its harmonics， or subharmonics. For example， take the second-stage blade of a hypothetical compressor. Its first flexural natural frequencyis calculated and found to be 200 Hz. From the Campbell diagram figure， it
.able 5-4
.haracteristics of Rotor Instabilities
Frequency .ype of Instability Onset Response .aused by
F rJ巾生 VibrGti n Unbalance Any speed .二 . Nonhomogeneous material Shaft misalignment Any speed .二 2. Driver and driven equipment misaligned
S巾l.-ExJit巾生 VibrGti n Hysteretic whirl . > .1 . . .1 Shrink fits and
.二 .5. built-up parts Hydrodynamic whirl . > 2.1 . . .5. Fluid film bearings (Oil Whip) and seals Aerodynamic whirl . > .1 .二 .1Compressor orturbine，
tip clearanceeffects， balance pistons Dry-friction whirl Any speed . 1二 n. Shaft in contact with stationary guide Entrained fluid .1 < . < 2..二 .1 Liquid or steam .5. < . < . entrapped in rotor
is apparent that a forcing frequency of12，000 rpm produced by operatingthe compressor at12，000 rpm will excite the 200-Hz first flexural frequency of the blade (200 Hz 60二12，000 rpm).Also， there are five inlet guide vanes ahead of the second-stage blade row. Operating the compressor at 2400 rpm will excite the 200 Hz natural frequency of the blade. (200 Hz 60二 5 2400 rpm.)
　
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